Remark 8 Observe that the smallness assumption made on \(T\)  in Theorem REF  is needed only to guarantee existence of a classical solution to the PDE (REF ).
By [1]}, this PDE admits a (Lipschitz-continuous) solution on \([0,T]\)  for all \(T>0\)  provided that the FBSDE solution satisfies
\(\mathbb {E}\big [|Y^{t,\chi ,\mathcal {L}(\chi )}_t - Y_t^{t,\chi ^{\prime },\mathcal {L}(\chi ^{\prime })} |^2 \big ]^{1/2} \le C\mathbb {E}[|\chi - \chi ^{\prime }|^2]^{1/2}\) 
